The fact that socialization is happening with the aid of a computer does not make it inherently more dangerous...
Yes it does. There is far more access to dangerous materials and dangerous people online than there is in person. While there's a slight advantage in a greater pool of potential victims to hide in the crowd of, the danger in the ability of predators of any stripe (not just sexual) to reach your kids from anywhere in the country or even the world. There's not as much ability for kids to tell what a "bad neighborhood" is online as in real life.
There's also less public shame for bad behavior and a greater tendency for people to act in herds of like-minded individuals. (See, e.g. the resurgence of white supremacist groups in the modern day or "thinspiration" sites.) You don't have to encounter people who disagree with you, unless you want to -- even if just to troll them. Witness comments section of any news or politics site.
[W]ithout the Interwebs this girl would still have been harassed, and we should be working to stop the harassment, not to stop the use of computers in harassment.
The harassment would have been completely different in tone and scale. Hiding behind a computer is quite different from doing something where witnesses who might disapprove would be present to act as a check or the much simpler one of being within arms reach. Witness Xbox Live, the domain of bullies who would be the bullied anywhere else.
Tools matter. There's a difference between two hotheaded boys getting in a fist fight and two armed hotheaded boys getting into a fight. The same is true of cyberbullying v. in-person bullying. People act differently in different environments, and online is more (and less) dangerous for certain types of behavior.